# Chapter 11 - SECTION 11.1(PAGE 597 R A ADAMS CALCULUS...

This preview shows pages 1–2. Sign up to view the full content.

SECTION 11.1 (PAGE 597) R. A. ADAMS: CALCULUS CHAPTER 11. VECTOR FUNCTIONS AND CURVES Section 11.1 Vector Functions of One Variable (page 597) 1. Position: r = i + t j Velocity: v = j Speed: v = 1 Acceleration : a = 0 Path: the line x = 1inthe xy -plane. 2. Position: r = t 2 i + k Velocity: v = 2 t i Speed: v = 2 | t | Acceleration : a = 2 i Path: the line z = 1, y = 0. 3. Position: r = t 2 j + t k Velocity: v = 2 t j + k Speed: v = 4 t 2 + 1 Acceleration : a = 2 j Path: the parabola y = z 2 in the plane x = 0. 4. Position: r = i + t j + t k Velocity: v = j + k Speed: v = 2 Acceleration : a = 0 Path: the straight line x = 1, y = z . 5. Position: r = t 2 i t 2 j + k Velocity: v = 2 t i 2 t j Speed: v = 2 2 t Acceleration: a = 2 i 2 j Path: the half-line x =− y 0, z = 1. 6. Position: r = t i + t 2 j + t 2 k Velocity: v = i + 2 t j + 2 t k Speed: v = 1 + 8 t 2 Acceleration: a = 2 j + 2 k Path: the parabola y = z = x 2 . 7. Position: r = a cos t i + a sin t j + ct k Velocity: v a sin t i + a cos t j + c k Speed: v = a 2 + c 2 Acceleration: a a cos t i a sin t j Path: a circular helix. 8. Position: r = a cos ω t i + b j + a sin ω t k Velocity: v a ω sin ω t i + a ω cos ω t k Speed: v =| a ω | Acceleration: a a ω 2 cos ω t i a ω 2 sin ω t k Path: the circle x 2 + z 2 = a 2 , y = b . 9. Position: r = 3 cos t i + 4 cos t j + 5 sin t k Velocity: v 3 sin t i 4 sin t j + 5 cos t k Speed: v = 9 sin 2 t + 16 sin 2 t + 25 cos 2 t = 5 Acceleration : a 3 cos t i 4 cos t j 5 sin t k r Path: the circle of intersection of the sphere x 2 + y 2 + z 2 = 25 and the plane 4 x = 3 y . 10. Position: r = 3 cos t i + 4 sin t j + t k Velocity: v 3 sin t i + 4 cos t j + k Speed: v = 9 sin 2 t + 16 cos 2 t + 1 = 10 + 7 cos 2 t Acceleration : a 3 cos t i 4 sin t j = t k r Path: a helix (spiral) wound around the elliptic cylinder ( x 2 / 9 ) + ( y 2 / 16 ) = 1. 11. Position: r = ae t i + be t j + ce t k Velocity and acceleration: v = a = r Speed: v = e t a 2 + b 2 + c 2 Path: the half-line x a = y b = z c > 0. 12. Position: r = at cos ω t i + sin ω t j + b ln t k Velocity: v = a ( cos ω t ω t sin ω t ) i + a ( sin ω t + ω t cos ω t ) j + ( b / t ) k Speed: v = ± a 2 ( 1 + ω 2 t 2 ) + ( b 2 / t 2 ) Acceleration: a a ω( 2 sin ω t + ω cos ω t ) i + a 2 cos ω t ω sin ω t ) j ( b / t 2 ) k Path: a spiral on the surface x 2 + y 2 = a 2 e z / b . 13. Position: r = e t cos ( e t ) i + e t sin ( e t ) j e t k Velocity: v ² e t cos ( e t ) + sin ( e t ) ³ i ² e t sin ( e t ) cos ( e t ) ³ j e t k Speed: v = 1 + e 2 t + e 2 t Acceleration: a = ² ( e t e t ) cos ( e t ) + sin ( e t ) ³ i + ² ( e t e t ) sin ( e t ) cos ( e t ) ³ j e t k Path: a spiral on the surface z ± x 2 + y 2 1. 14. Position: r = a cos t sin t i + a sin 2 t j + a cos t k = a 2 sin 2 t i + a 2 ² 1 cos 2 t ³ j + a cos t k Velocity: v = a cos 2 t i + a sin 2 t j a sin t k Speed: v = a 1 + sin 2 t Acceleration: a 2 a sin 2 t i + 2 a cos 2 t j a cos t k Path: the path lies on the sphere x 2 + y 2 + z 2 = a 2 ,on the surface deﬁned in terms of spherical polar coordinates by φ = θ , on the circular cylinder x 2 + y 2 = ay , and on the parabolic cylinder + z 2 = a 2 . Any two of these surfaces serve to pin down the shape of the path.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 28

Chapter 11 - SECTION 11.1(PAGE 597 R A ADAMS CALCULUS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online